Gilles A. Francfort

Revisiting brittle fracture as an energy minimization problem

Abstract A variational model of quasistatic crack evolution is proposed. Although close in spirit to Griffith’s theory of brittle fracture, the proposed model however frees itself of the usual constraints of that theory : a preexisting crack and a well-defined crack path. In contrast, crack initiation as well as crack path can be quantified, as demonstrated on explicitly computable examples. Furthermore the model lends itself to numerical implementation in more complex settings.

The Variational Approach to Fracture

1 Introduction 2 Going variational 2.1 Griffiths theory 2.2 The 1-homogeneous case - A variational equivalence 2.3 Smoothness - The soft belly of Griffiths formulation 2.4 The non 1-homogeneous case - A discrete variational evolution 2.5 Functional framework - A weak variational evolution 2.6 Cohesiveness and the variational evolution 3 Stationarity versus local or global minimality - A comparison 3.1 1d traction 3.1.1 The Griffith case - Soft device 3.1.2 The Griffith case - Hard device 3.1.3 Cohesive case - Soft device 3.1.4 Cohesive case - Hard device 3.2 A tearing experiment 4 Initiation 4.1 Initiation - The Griffith case 4.1.1 Initiation - The Griffith case - Global minimality 4.1.2 Initiation - The Griffith case - Local minimality 4.2 Initiation - The cohesive case 4.2.1 Initiation - The cohesive 1d case - Stationarity 4.2.2 Initiation - The cohesive 3d case - Stationarity 4.2.3 Initiation - The cohesive case - Global minimality 5 Irreversibility 5.1 Irreversibility - The Griffith case - Well-posedness of the variational evolution 5.1.1 Irreversibility - The Griffith case - Discrete evolution 5.1.2 Irreversibility - The Griffith case - Global minimality in the limit 5.1.3 Irreversibility - The Griffith case - Energy balance in the limit 5.1.4 Irreversibility - The Griffith case - The time-continuous evolution 5.2 Irreversibility - The cohesive case 6 Path 7 Griffith vs. Barenblatt 8 Numerics and Griffith 8.1 Numerical approximation of the energy 8.1.1 The first time step 8.1.2 Quasi-static evolution 8.2 Minimization algorithm 8.2.1 The alternate minimization algorithm 8.2.2 The backtracking algorithm 8.3 Numerical experiments 8.3.1 The 1D traction (hard device) 8.3.2 The Tearing experiment 8.3.3 Revisiting the 2D traction experiment on a fiber reinforced matrix 9 Fatigue 9.1 Peeling Evolution 9.2 The limitfatigue law when d tends to 0 9.3 A variational formulation for fatigue 9.3.1 Peeling revisited 9.3.2 Generalization Appendix Glossary References.

Numerical experiments in revisited brittle fracture

Abstract The numerical implementation of the model of brittle fracture developed in Francfort and Marigo (1998. J. Mech. Phys. Solids 46 (8), 1319–1342) is presented. Various computational methods based on variational approximations of the original functional are proposed. They are tested on several antiplanar and planar examples that are beyond the reach of the classical computational tools of fracture mechanics.

Homogenization and Linear Thermoelasticity

We study homogenization of linear dynamic thermoelasticity with rapidly varying coefficients, using a semigroup approach. The resulting homogenized problem exhibits an unusual change in initial temperature.A formal asymptotic analysis predicts fast time oscillations in the temperature field. These oscillations explain the temperature shift, and show that, for our problem, weak convergence in time is the best convergence that one can obtain.

Erratum: study of a doubly nonlinear hear equation with no growth assumptions on the parabolic term

A doubly nonlinear equation with no growth assumptions on the parabolic term or on the heat flux is studied. Two existence and comparison results are established under different assumptions on the data. The technique uses truncation-penalization of the energy and energy estimates through convex conjugate functions.

3D-2D ASYMPTOTIC ANALYSIS OF AN OPTIMAL DESIGN PROBLEM FOR THIN FILMS

Abstract The Gamma-limit of a rescaled version of an optimal material distribution problem for a cylindrical two-phase elastic mixture in a thin three-dimensional domain is explicitly computed. Its limit is a two-dimensional optimal design problem on the cross-section of the thin domain; it involves optimal energy bounds on two-dimensional mixtures of a related two-phase bulk material. Thus, it is shown in essence that 3D-2D asymptotics and optimal design commute from a variational standpoint.

Revisiting Energy Release Rates in Brittle Fracture

We revisit in a 2d setting the notion of energy release rate, which plays a pivotal role in brittle fracture. Through a blow-up method, we extend that notion to crack patterns which are merely closed sets connected to the crack tip. As an application, we demonstrate that, modulo a simple meta-stability principle, a moving crack cannot generically kink while growing continuously in time. This last result potentially renders obsolete in our opinion a longstanding debate in fracture mechanics on the correct criterion for kinking.

Quasi-static Evolution in Nonassociative Plasticity: The Cap Model

Non-associative elasto-plasticity is the working model of plasticity for soil and rocks mechanics. Yet, it is usually viewed as non-variational. In this work, we prove a contrario the existence of a variational evolution for such a model under a natural capping assumption on the hydrostatic stresses and a less natural mollication of the stress admissibility constraint. The obtained elasto-plastic evolution is expressed for times that are conveniently rescaled.

Existence of minimizers for non-quasiconvex functionals arising in optimal design

This paper investigates the existence of minimizers for the so-called Kohn-Strang functional with affine boundary conditions. Such a functional, which arises in optimal shape design problems in electrostatics, is not quasi-convex, and therefore existence of minimizers is, in general, guaranteed only for its quasi-convex envelope. Such a quasi-convexification has been computed in two space dimensions in [11]. Recently, necessary and sufficient conditions on the affine boundary conditions for existence of minimizers for the Kohn-Strang functional have been derived in two space dimensions in [7]. We generalize these previous results for arbitrary space dimensions. Our method relies on the homogenization approach for relaxing optimal design problems. We also generalize our results to some variants of the Kohn-Strang functional.

Fourth-order moments of nonnegative measures on S2 and applications

This paper is mainly devoted to a study of the fifteen-dimensional set of all fourth-order moments of a nonnegative Radon (Borel regular) measure on S 2. We seek a complete characterization of the set, a closed cone, with the help of the Hilbert decomposition theorem for nonnegative polynomials of degree four in two variables. Special emphasis is placed on the boundary of this set which is shown to be generated by atomic measures made of five Dirac masses or fewer, which are located on the intersection of S 2 with the zero set of a quadratic form. As a consequence, every point of the moment set is generated by atomic measures made of six Dirac masses or less. This study may be viewed as a contribution to the moment problem; note that the analogous two-dimensional case has been analyzed in AVELLANEDA & MILTON [AM]. Potential applications for this result are however manifold, especially in the field of homogenization, which is of especial interest to us. In the setting of linearized elasticity, effective properties of fine mixtures of two phases are investigated. Specifically, the goal is to analyze the coefficients of the linear second-order elliptic system satisfied by the weak limit u of the solution fields W to a sequence of elasticity problems of the form